Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups The number of facets of three-dimensional Dirichlet stereohedra Francisco Santos (w. D. Bochis, P. Sabariego), U. Cantabria. ERC Workshop “Delaunay Geometry”; Berlin, oct 2013 F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have? F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have? Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have? Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). Dirichlet stereohedron: the Voronoi region of a point p ∈ S with respect to an orbit S of G. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have? Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). Dirichlet stereohedron: the Voronoi region of a point p ∈ S with respect to an orbit S of G. P F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have? Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). Dirichlet stereohedron: the Voronoi region of a point p ∈ S with respect to an orbit S of G. P F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. Engel (1980) found Dirichlet stereohedra with 38 facets for the cubic group I 4132 F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. Engel (1980) found Dirichlet stereohedra with 38 facets for the cubic group I 4132 Our goal: lower the gap between 38 and 390 (by improving the upper bound). F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. Engel (1980) found Dirichlet stereohedra with 38 facets for the cubic group I 4132 Our goal: lower the gap between 38 and 390 (by improving the upper bound). Our global upper bound: 92. F. Santos Number of facets of 3-d Dirichlet stereohedra W 967 a va a o x. ov, a o o, a g o a yg g av 1, 2 o 3 va a aa, a a o o , o v a o o o , va. y g a o a a o vaay o aa v o go a a, a a go o o o a y o . a o g 12 a 13, a o a o o go y ao a a o a o a . aga x a gv a ay o, a a, vy o o a ay y o o o o va o . a o o o a o a ov a oo o og o o a vy Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups a o oga o. o a intro. a o o aa, v gao, a v Engel’s o stereohedra, a a o xy . F. Santos Number of facets of 3-d Dirichlet stereohedra 13 a 13. x a o y o oa o y g 1980. g ay v gao o o y o aa o o a, g a o o aa va « 0.0158. a a o aa va 0.20 x 0.23 a 0.14 0.20, o g a o a ag a 0.214 x 0.220, 0.145 . y o — p, a o a go o a, a o , o v, a p a a o g y a o a Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra? F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra? “The total absence of methods usable in more general situations” (Gr¨unbaum-Shephard, 1980). F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra? “The total absence of methods usable in more general situations” (Gr¨unbaum-Shephard, 1980). Even if “there seems to be no grounds to assume that all stereohedra are combinatorially equivalent to Dirichlet stereohedra” (op. cit.) F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra? “The total absence of methods usable in more general situations” (Gr¨unbaum-Shephard, 1980). Even if “there seems to be no grounds to assume that all stereohedra are combinatorially equivalent to Dirichlet stereohedra” (op. cit.) Also, there is an algorithm to compute all Dirichlet stereohedra in a given dimension “if carried out with sufficient perseverance”, (op. cit.). The same is not true for arbitrary stereohedra. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo Our method combines general principles with case-by-case studies. F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: Groups that contain reflection planes (the reptilarium). F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: Groups that contain reflection planes (the reptilarium). Rest of non-cubic groups (fish and birds). F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro.